Norm conditions for uniform algebra isomorphisms

• Autorzy: Aaron Luttman , Scott Lambert
• Języki publikacji: EN

Abstrakty

EN

In recent years much work has been done analyzing maps, not assumed to be linear, between uniform algebras that preserve the norm, spectrum, or subsets of the spectra of algebra elements, and it is shown that such maps must be linear and/or multiplicative. Letting A and B be uniform algebras on compact Hausdorff spaces X and Y, respectively, it is shown here that if λ ∈ ℂ / {0} and T: A → B is a surjective map, not assumed to be linear, satisfying $$ \left\| {T(f)T(g) + \lambda } \right\| = \left\| {fg + \lambda } \right\|\forall f,g \in A, $$ then T is an ℝ-linear isometry and there exist an idempotent e ∈ B, a function κ ∈ B with κ 2 = 1, and an isometric algebra isomorphism $$ \tilde T:{\rm A} \to Be \oplus \bar B(1 - e) $$ such that $$ T(f) = \kappa \left( {\tilde T(f)e + \gamma \overline {\tilde T(f)} (1 - e)} \right) $$ for all f ∈ A, where γ = λ / |λ|. Moreover, if T is unital, i.e. T(1) = 1, then T(i) = i implies that T is an isometric algebra isomorphism whereas T(i) = −i implies that T is a conjugate-isomorphism.

Słowa kluczowe

EN

uniform algebras; peripheral spectrum; isometric algebra isomorphism

Kategorie tematyczne

• 46J20: Ideals, maximal ideals, boundaries
• 46J10: Banach algebras of continuous functions, function algebras
• 46H40: Automatic continuity

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2008
• Tom: 6
• Numer: 2
• Strony: 272-280

Daty

wydano

2008-06-01

online

2008-04-15

Twórcy

autor

Aaron Luttman

• Division of Science and Mathematics

autor

Scott Lambert

• University of Montana

Bibliografia

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Identyfikatory

DOI

10.2478/s11533-008-0016-x